Use Descartes' Rule of Signs to determine the possible numbers of positive and negative real zeros. $$ g(x)=9 x^{3}+8 x^{2}+9 x-6 $$ If there is more than one possibility, separate them with commas. (a) Possible number(s) of positive real zeros: (b) Possible number(s) of negative real zeros:

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**Step 1. Count the sign changes for positive real zeros**

For positive $$x$$, consider the polynomial:
$$
g(x)=9x^3+8x^2+9x-6
$$
Examine the coefficients: $$9$$, $$8$$, $$9$$, $$-6$$.

- From $$9$$ to $$8$$: no sign change (both are positive).
- From $$8$$ to $$9$$: no sign change (both are positive).
- From $$9$$ to $$-6$$: sign change (positive to negative).

Total sign changes: $$1$$.

Thus by Descartes' Rule of Signs, the possible number of positive real zeros is:
$$
1
$$

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**Step 2. Count the sign changes for negative real zeros**

To determine the number of negative real zeros, substitute $$-x$$ for $$x$$ in the polynomial:
$$
g(-x)=9(-x)^3+8(-x)^2+9(-x)-6
$$
Simplify each term:
$$
g(-x)=-9x^3+8x^2-9x-6
$$

Now, examine the coefficients: $$-9$$, $$8$$, $$-9$$, $$-6$$.

- From $$-9$$ to $$8$$: sign change (negative to positive) – change $$\#1$$.
- From $$8$$ to $$-9$$: sign change (positive to negative) – change $$\#2$$.
- From $$-9$$ to $$-6$$: no sign change (both negative).

Total sign changes: $$2$$.

Thus, by Descartes' Rule of Signs, the possible number of negative real zeros is either:
$$
2 \quad \text{or} \quad 0
$$

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**Final Answers:**

(a) Possible number(s) of positive real zeros: $$1$$

(b) Possible number(s) of negative real zeros: $$2, 0$$
(a) Possible number(s) of positive real zeros: 1 (b) Possible number(s) of negative real zeros: 2, 0

Use Descartes’ Rule of Signs to determine the possible numbers of positive and negative real zeros.

If there is more than one possibility, separate them with commas.
(a) Possible number(s) of positive real zeros:
(b) Possible number(s) of negative real zeros:

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Use Descartes’ Rule of Signs to determine the possible numbers of positive and negative real zeros. If there is more than one possibility, separate them with commas. (a) Possible number(s) of positive real zeros: (b) Possible number(s) of negative real zeros: